All of the 4th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$2.50$ each for students, and the group paid $$30.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$20.00$ each for teachers and $$11.50$ each for students, and the group paid $$129.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+2.5y = 30}$ ${20x+11.5y = 129}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-10y = -120}$ ${20x+11.5y = 129}$ Add the top and bottom equations together. $ 1.5y = 9 $ $ y = \dfrac{9}{1.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+2.5y = 30}$ to find $x$ ${5x + 2.5}{(6)}{= 30}$ $5x+15 = 30$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {20x+11.5y = 129}$ and get the same answer for $x$ ${20x + 11.5}{(6)}{= 129}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.